Hidden Math Toolkit Could Reshape How We Model Evolution

Every living species carries within it a record of millions of years of change: genes duplicated and lost, populations split and merged, body sizes shifted as environments transformed. Evolutionary biologists have spent decades building mathematical models to decode that record, yet many of the tools they rely on remain surprisingly limited. A new perspective article published in Frontiers in Systems Biology argues that a powerful and largely untapped class of probability models, known as Markov modulated models from the theory of matrix-analytic methods, could dramatically expand what researchers can learn from evolutionary data. The article, written by Małgorzata M. O’Reilly and Barbara R. Holland of the University of Tasmania, makes the case that unlocking this potential will require a deliberate meeting of minds between applied probabilists and evolutionary biologists.

The central idea behind a Markov modulated model is deceptively simple. Imagine a complex system whose behaviour changes over time in ways that are hard to predict directly. Rather than modelling that system in isolation, the model introduces an underlying Markov environment, called the phase, which drives or modulates the evolution of the quantity of interest, called the level. The phase acts like an invisible switchboard: as it moves between states, it changes the rules governing how the level fluctuates. In evolutionary terms, the level might represent the number of genes in a gene family or the body mass of a species, while the phase might capture an environmental condition or a discrete trait such as egg-laying versus live birth that alters how those quantities change.

O’Reilly and Holland focus on two fundamental classes of these models. The first is the quasi-birth-and-death process, or QBD, in which the level variable is discrete. A QBD is a continuous-time Markov chain with a skip-free structure, meaning transitions are only possible between neighbouring levels. This restriction produces a generator matrix partitioned into small, finite-size blocks containing transition rates between adjacent levels, which in turn enables efficient numerical analysis through fast algorithms. In an evolutionary context, the level of a QBD could track the number of individuals carrying a particular allele in a population, while the phase represents environments in which that allele experiences stronger or weaker selection. The second class is the stochastic fluid model, or SFM, in which the level is a continuous quantity whose rate of increase or decrease is modulated by a finite-state continuous-time Markov chain. Here the level might describe a continuous trait such as body mass, rising, falling, or holding steady depending on the phase. According to the authors, SFMs have not yet been applied in an evolutionary context, but they could offer an alternative to existing methods such as SURFACE, a generalisation of the Ornstein-Uhlenbeck process used to detect regime shifts in trait evolution along a phylogeny.

One of the key advantages of these models, the authors emphasise, is computational. Matrix-analytic methods, pioneered by Marcel Neuts in the late 1970s and early 1980s, replace complicated and tedious calculations with matrix manipulations, yielding closed-form or recursive expressions for important quantities, including likelihoods. Parameters can then be estimated through numerical optimisation of the likelihood or informed by prior biological knowledge. The models also translate naturally into simulations, producing insightful numerical outputs that can be compared against real data. This combination of analytical tractability and simulation flexibility is rare in evolutionary modelling, where researchers often must choose between mathematically convenient but biologically unrealistic models and realistic but intractable ones.

A particularly powerful feature of Markov modulated models is their ability to represent a wide range of distributions for the time spent in a particular state, going well beyond the exponential waiting times that constrain many classical evolutionary models. This flexibility comes from phase-type distributions, which record the time until absorption in an absorbing Markov chain. Phase-type distributions can approximate any positive-valued distribution and can be fitted to empirical data using established tools. The authors point out that several existing evolutionary models are already special cases of phase-type models, sometimes without researchers realising it. The covarion model of DNA substitution, for instance, can be viewed as a mechanistic phase-type model in which the phase corresponds to a site being able or unable to accept a substitution. Similarly, recent work has shown that phase-type distributed times to speciation give superior fits to empirical species trees in terms of their balance, and that important quantities in coalescent models can often be expressed in phase-type terms, providing a unified theory for a range of population genetics models.

To demonstrate the practical potential of these ideas, the authors present a general tree likelihood formula applicable to a wide range of Markov modulated models. Suppose the evolution of some characters within a phylogenetic tree is modelled using a QBD or an SFM, for example a gene family evolving within a species tree or a trait evolving along its branches. By generalising results previously established for QBDs and Markovian binary trees, the likelihood of observing the tree can be evaluated with a recursive algorithm whose structure is analogous to the classical pruning algorithm of Felsenstein. The algorithm works from the tips of the tree toward the root, combining conditional probability or density vectors from subtrees using Hadamard products, and it can be modified to incorporate observed states of the model at points within the tree. This means that maximum likelihood, the well-established workhorse of phylogenetic inference, can be brought to bear on models far richer than those currently in routine use.

The authors also identify where existing evolutionary models fit within this broader framework. Markovian binary trees, a class of models with a tree-like structure that falls within the QBD category, have been shown to encompass several later-introduced evolutionary models as special cases. This means the existing theory of Markovian binary trees and QBDs can be readily applied to derive many useful quantities of interest, such as the probability of a species becoming extinct, the probability of survival, or the likelihood of observing a species-gene tree, considerably extending current analyses through efficient matrix-analytic techniques involving differential equations and recursive algorithms written in convenient matrix form.

Significant challenges remain, however, and the authors are candid about them. The number of alternative tree topologies grows explosively with the number of tips, following the sequence of Euler numbers, so reconstructing a tree from tip data alone is computationally daunting. The authors propose tackling this with dynamic programming, reconstructing the tree from the tips to the root while storing intermediate computations, and they suggest that approximate dynamic programming techniques, which are designed for large-scale optimisation problems, could make the approach feasible. Another major gap concerns time-inhomogeneity. Most current matrix-analytic theory assumes time-homogeneous parameters, yet evolutionary processes are rarely constant through time. Preliminary work on time-inhomogeneous cyclic QBDs and stochastic fluid models exists, and the authors see strong potential for adapting these ideas to evolutionary problems, for example to model cyclical mutation modalities. Methods to handle delays in response to environmental stimuli have not yet been developed in the Markov modulated literature at all. Multi-dimensional level variables offer yet another frontier, potentially enabling analysis of environmentally-driven evolution of species in more realistic detail.

Practical software is already in place to support this agenda. The BuTools library provides functions for analysing QBDs and other Markovian models, including phase-type distributions, in Matlab, Mathematica and Python. SMCSolver computes stationary distributions of QBDs, and MAMSolver implements matrix-analytic algorithms in Matlab and C++. These tools lower the barrier for evolutionary biologists willing to experiment with the framework, even before new theory is developed. The authors also note that the likelihood algorithm they present, which has been applied in the context of QBDs and Markovian binary trees for tree likelihood evaluation and reconciliation problems, could be extended to stochastic fluid models and to time-inhomogeneous Markov modulated models, widening the class of evolutionary questions that can be addressed.

Ultimately, the message of the perspective is one of untapped opportunity. A rich, well-developed body of probability theory already exists, honed over decades in fields such as queueing theory and performance evaluation, and much of it has never been pointed at evolutionary questions. Conversely, deep knowledge of evolutionary processes is needed to formulate theoretical problems worth solving, to identify which biological mechanisms should be encoded in the phase variable, and to judge which model refinements matter most. The authors argue that only close multidisciplinary collaboration between applied probabilists and evolutionary biologists can unlock the full application potential of these models. If that collaboration takes hold, the hidden machinery of matrix-analytic methods may soon move from the margins of applied probability into the mainstream of evolutionary research, offering new ways to read the history of life written in genes, traits, and trees.

Subject of Research: Matrix-analytic Markov modulated models and their application potential for evolutionary biology

Article Title: On matrix-analytic Markov modulated models for evolution: application potential and future directions required to advance the field

Article References: O’Reilly, M. M., & Holland, B. R. (2026). On matrix-analytic Markov modulated models for evolution: application potential and future directions required to advance the field. Frontiers in Systems Biology, 6, Article 1906462. https://doi.org/10.3389/fsysb.2026.1906462

Image Credits: AI Generated

DOI: 10.3389/fsysb.2026.1906462

Keywords: matrix-analytic methods, Markov modulated models, quasi-birth-and-death processes, stochastic fluid models, phase-type distributions, phylogenetic inference, evolutionary biology, gene family evolution, likelihood, computational probability, matrix-analytic, Markov